4-7 The Quadratic Formula

Objectives

To solve quadratic equations using the Quadratic Formula

To determine the number of solutions by using the discriminant

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Another way to solve a quadratic equation a x 2 + b x + c = 0 eh , x squared , plus b x plus c equals 0  is by completing a square and then factoring.

Essential Understanding You can solve a quadratic equation a x 2 + b x + c = 0 eh , x squared , plus b x plus c equals 0  in more than one way. In general, you can find a formula that gives values of x in terms of a, b, and c.

Here's how to solve a x 2 + b x + c = 0 eh , x squared , plus b x plus c equals 0  to get the Quadratic Formula.

a x 2 + b x + c = 0   x 2 + b a x + c a = 0 Divide each side by  a . x 2 + b a x = − c a Rewrite so all terms containing  x  are on one side . x 2 + b a x + ( b 2 a ) 2 = ( b 2 a ) 2 − c a Complete the square . ( x + b 2 a ) 2 = b 2 − 4 a c 4 a 2 Factor the perfect square trinomial . Also, simplify . x + b 2 a = ± b 2 − 4 a c 4 a 2 Find square roots . x = − b 2 a ± b 2 − 4 a c 2 a Solve for  x .  Also, simplify the radical . x = − b ± b 2 − 4 a c 2 a Simplify . table with 8 rows and 3 columns , row1 column 1 , eh , x squared , plus b x plus c , column 2 equals 0 , column 3 , row2 column 1 , x squared , plus , b over eh , x plus , c over eh , column 2 equals 0 , column 3 cap divideeachsideby . eh . , row3 column 1 , x squared , plus , b over eh , x , column 2 equals negative , c over eh , column 3 cap rewritesoalltermscontaining . x . areononeside . . , row4 column 1 , x squared , plus , b over eh , x plus . open , fraction b , over 2 eh end fraction , close squared , column 2 equals . open , fraction b , over 2 eh end fraction , close squared . minus , c over eh , column 3 cap completethesquare . . , row5 column 1 , open . x plus , fraction b , over 2 eh end fraction . close squared , column 2 equals . fraction b squared , minus 4 eh c , over 4 , eh squared end fraction , column 3 cap factortheperfectsquaretrinomial . .cap alsocommasimplify . . , row6 column 1 , x plus , fraction b , over 2 eh end fraction , column 2 equals plus minus . square root of fraction b squared , minus 4 eh c , over 4 , eh squared end fraction end root , column 3 cap findsquareroots . . , row7 column 1 , x , column 2 equals negative , fraction b , over 2 eh end fraction , plus minus . fraction square root of b squared , minus 4 eh c end root , over 2 eh end fraction , column 3 cap solvefor . x . . cap alsocommasimplifytheradical . . , row8 column 1 , x , column 2 equals . fraction negative b plus minus . square root of b squared , minus 4 eh c end root , over 2 eh end fraction , column 3 cap simplify , . , end table

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